Properties

Label 325.2.x.b
Level $325$
Weight $2$
Character orbit 325.x
Analytic conductor $2.595$
Analytic rank $0$
Dimension $20$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [325,2,Mod(7,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 325.x (of order \(12\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.59513806569\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(5\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - \beta_{19} + \beta_{18} + \cdots + \beta_{2}) q^{3}+ \cdots + ( - \beta_{18} + \beta_{17} + 2 \beta_{16} + \cdots + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - \beta_{19} + \beta_{18} + \cdots + \beta_{2}) q^{3}+ \cdots + (3 \beta_{19} - 2 \beta_{18} + \cdots + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 6 q^{2} + 2 q^{3} + 6 q^{4} - 8 q^{6} + 2 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 6 q^{2} + 2 q^{3} + 6 q^{4} - 8 q^{6} + 2 q^{7} + 12 q^{9} - 16 q^{11} + 24 q^{12} + 4 q^{13} - 2 q^{16} - 4 q^{17} - 20 q^{19} + 4 q^{21} - 16 q^{22} + 10 q^{23} + 32 q^{24} - 24 q^{26} - 4 q^{27} - 18 q^{28} - 48 q^{32} - 18 q^{33} + 2 q^{34} + 36 q^{36} + 4 q^{37} + 8 q^{38} + 4 q^{39} + 10 q^{41} - 40 q^{42} - 10 q^{43} - 36 q^{44} + 4 q^{46} + 40 q^{47} + 56 q^{48} + 18 q^{49} + 30 q^{52} + 10 q^{53} - 48 q^{54} - 16 q^{59} - 16 q^{61} + 44 q^{62} + 36 q^{63} + 20 q^{64} - 32 q^{66} - 18 q^{67} - 22 q^{68} - 16 q^{69} - 16 q^{71} - 4 q^{72} + 18 q^{74} - 64 q^{76} + 28 q^{77} - 68 q^{78} - 14 q^{81} - 56 q^{82} - 48 q^{83} - 40 q^{84} + 60 q^{86} + 34 q^{87} - 82 q^{88} - 6 q^{89} + 8 q^{91} + 8 q^{92} - 32 q^{93} - 48 q^{94} + 56 q^{96} - 66 q^{97} + 30 q^{98} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 26 x^{18} + 279 x^{16} + 1604 x^{14} + 5353 x^{12} + 10466 x^{10} + 11441 x^{8} + 6176 x^{6} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 20 \nu^{18} - 389 \nu^{16} - 2695 \nu^{14} - 7125 \nu^{12} + 1214 \nu^{10} + 39860 \nu^{8} + \cdots - 409 ) / 5992 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 20 \nu^{18} + 389 \nu^{16} + 2695 \nu^{14} + 7125 \nu^{12} - 1214 \nu^{10} - 39860 \nu^{8} + \cdots + 409 ) / 5992 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 69 \nu^{18} - 1647 \nu^{16} - 15798 \nu^{14} - 78322 \nu^{12} - 214723 \nu^{10} - 324081 \nu^{8} + \cdots - 539 ) / 1712 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 321 \nu^{19} - 483 \nu^{18} + 9951 \nu^{17} - 11529 \nu^{16} + 127330 \nu^{15} - 110586 \nu^{14} + \cdots - 3773 ) / 23968 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 700 \nu^{19} + 617 \nu^{18} + 17360 \nu^{17} + 16607 \nu^{16} + 174468 \nu^{15} + 185192 \nu^{14} + \cdots + 26961 ) / 11984 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 409 \nu^{19} - 10614 \nu^{17} - 113722 \nu^{15} - 653341 \nu^{13} - 2182252 \nu^{11} - 4281808 \nu^{9} + \cdots - 2996 ) / 5992 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1635 \nu^{19} - 309 \nu^{18} + 41725 \nu^{17} - 7171 \nu^{16} + 436590 \nu^{15} - 66542 \nu^{14} + \cdots - 30699 ) / 23968 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 700 \nu^{19} - 617 \nu^{18} + 17360 \nu^{17} - 16607 \nu^{16} + 174468 \nu^{15} - 185192 \nu^{14} + \cdots - 26961 ) / 11984 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 419 \nu^{19} + 4593 \nu^{18} - 12681 \nu^{17} + 118807 \nu^{16} - 158886 \nu^{15} + 1266230 \nu^{14} + \cdots + 83703 ) / 23968 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3773 \nu^{19} - 565 \nu^{18} - 97615 \nu^{17} - 11551 \nu^{16} - 1041138 \nu^{15} - 83062 \nu^{14} + \cdots + 81509 ) / 23968 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 3773 \nu^{19} + 321 \nu^{18} + 97615 \nu^{17} + 9951 \nu^{16} + 1041138 \nu^{15} + 127330 \nu^{14} + \cdots + 39483 ) / 23968 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 3773 \nu^{19} - 321 \nu^{18} + 97615 \nu^{17} - 9951 \nu^{16} + 1041138 \nu^{15} - 127330 \nu^{14} + \cdots - 39483 ) / 23968 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2753 \nu^{19} - 867 \nu^{18} - 71035 \nu^{17} - 22593 \nu^{16} - 754642 \nu^{15} - 243222 \nu^{14} + \cdots - 15221 ) / 11984 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 2753 \nu^{19} - 867 \nu^{18} + 71035 \nu^{17} - 22593 \nu^{16} + 754642 \nu^{15} - 243222 \nu^{14} + \cdots - 15221 ) / 11984 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 6965 \nu^{19} + 1863 \nu^{18} - 180971 \nu^{17} + 49605 \nu^{16} - 1940134 \nu^{15} + 545958 \nu^{14} + \cdots + 1285 ) / 23968 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 8297 \nu^{19} - 175 \nu^{18} + 216915 \nu^{17} - 2093 \nu^{16} + 2342074 \nu^{15} + 9562 \nu^{14} + \cdots + 23947 ) / 23968 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 4979 \nu^{19} - 129835 \nu^{17} - 1398012 \nu^{15} - 8068552 \nu^{13} - 27041293 \nu^{11} + \cdots + 11984 ) / 11984 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 10417 \nu^{19} - 483 \nu^{18} + 268635 \nu^{17} - 11529 \nu^{16} + 2853942 \nu^{15} - 110586 \nu^{14} + \cdots + 20195 ) / 23968 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 16083 \nu^{19} - 483 \nu^{18} - 419809 \nu^{17} - 11529 \nu^{16} - 4525598 \nu^{15} + \cdots - 15757 ) / 23968 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{12} + \beta_{10} - \beta_{3} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{19} + 2 \beta_{18} - \beta_{13} - 3 \beta_{12} - 2 \beta_{11} - 2 \beta_{10} - \beta_{9} + \cdots - 7 \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{19} - \beta_{18} + \beta_{17} + 2 \beta_{16} - 2 \beta_{15} + 3 \beta_{14} - 6 \beta_{12} + \cdots + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 9 \beta_{19} - 17 \beta_{18} - \beta_{17} - 3 \beta_{14} + 11 \beta_{13} + 26 \beta_{12} + 17 \beta_{11} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 13 \beta_{19} + 13 \beta_{18} - 13 \beta_{17} - 26 \beta_{16} + 26 \beta_{15} - 32 \beta_{14} + \cdots - 39 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 66 \beta_{19} + 122 \beta_{18} + 14 \beta_{17} + 37 \beta_{14} - 89 \beta_{13} - 189 \beta_{12} + \cdots + 21 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 120 \beta_{19} - 116 \beta_{18} + 116 \beta_{17} + 232 \beta_{16} - 240 \beta_{15} + 260 \beta_{14} + \cdots + 213 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 462 \beta_{19} - 840 \beta_{18} - 136 \beta_{17} - 332 \beta_{14} + 657 \beta_{13} + 1306 \beta_{12} + \cdots - 170 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 974 \beta_{19} + 914 \beta_{18} - 914 \beta_{17} - 1828 \beta_{16} + 1948 \beta_{15} - 1941 \beta_{14} + \cdots - 1255 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 3210 \beta_{19} + 5733 \beta_{18} + 1135 \beta_{17} + 2648 \beta_{14} - 4690 \beta_{13} - 8890 \beta_{12} + \cdots + 1267 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7438 \beta_{19} - 6832 \beta_{18} + 6832 \beta_{17} + 13664 \beta_{16} - 14876 \beta_{15} + 14032 \beta_{14} + \cdots + 7797 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 22356 \beta_{19} - 39188 \beta_{18} - 8780 \beta_{17} - 19984 \beta_{14} + 33058 \beta_{13} + \cdots - 9130 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 54982 \beta_{19} + 49778 \beta_{18} - 49778 \beta_{17} - 99556 \beta_{16} + 109964 \beta_{15} + \cdots - 50290 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 156271 \beta_{19} + 269080 \beta_{18} + 65190 \beta_{17} + 146494 \beta_{14} - 231975 \beta_{13} + \cdots + 64782 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 398897 \beta_{19} - 357747 \beta_{18} + 357747 \beta_{17} + 715494 \beta_{16} - 797794 \beta_{15} + \cdots + 333034 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( - 1095413 \beta_{19} - 1856511 \beta_{18} - 472963 \beta_{17} - 1056441 \beta_{14} + 1625819 \beta_{13} + \cdots - 456258 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 2861437 \beta_{19} + 2550751 \beta_{18} - 2550751 \beta_{17} - 5101502 \beta_{16} + 5722874 \beta_{15} + \cdots - 2246355 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( 7691530 \beta_{19} + 12862874 \beta_{18} + 3385870 \beta_{17} + 7545155 \beta_{14} - 11394317 \beta_{13} + \cdots + 3202109 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-\beta_{11} - \beta_{12}\) \(\beta_{12}\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
1.51805i
0.274809i
0.493902i
1.83163i
2.64975i
1.51805i
0.274809i
0.493902i
1.83163i
2.64975i
2.08794i
1.58474i
0.131303i
1.02262i
2.25081i
2.08794i
1.58474i
0.131303i
1.02262i
2.25081i
−1.31467 0.759023i −0.175069 0.653367i 0.152233 + 0.263675i 0 −0.265763 + 0.991842i 1.29744 + 2.24723i 2.57390i 2.20184 1.27123i 0
7.2 0.237991 + 0.137404i −0.611610 2.28256i −0.962240 1.66665i 0 0.168076 0.627267i 0.193052 + 0.334376i 1.07848i −2.23793 + 1.29207i 0
7.3 0.427732 + 0.246951i 0.243392 + 0.908353i −0.878030 1.52079i 0 −0.120212 + 0.448637i −1.83775 3.18307i 1.85513i 1.83221 1.05783i 0
7.4 1.58624 + 0.915816i 0.512942 + 1.91432i 0.677439 + 1.17336i 0 −0.939520 + 3.50634i 1.76945 + 3.06478i 1.18163i −0.803451 + 0.463873i 0
7.5 2.29475 + 1.32488i −0.335680 1.25278i 2.51060 + 4.34849i 0 0.889471 3.31955i −0.0561740 0.0972962i 8.00544i 1.14131 0.658935i 0
93.1 −1.31467 + 0.759023i −0.175069 + 0.653367i 0.152233 0.263675i 0 −0.265763 0.991842i 1.29744 2.24723i 2.57390i 2.20184 + 1.27123i 0
93.2 0.237991 0.137404i −0.611610 + 2.28256i −0.962240 + 1.66665i 0 0.168076 + 0.627267i 0.193052 0.334376i 1.07848i −2.23793 1.29207i 0
93.3 0.427732 0.246951i 0.243392 0.908353i −0.878030 + 1.52079i 0 −0.120212 0.448637i −1.83775 + 3.18307i 1.85513i 1.83221 + 1.05783i 0
93.4 1.58624 0.915816i 0.512942 1.91432i 0.677439 1.17336i 0 −0.939520 3.50634i 1.76945 3.06478i 1.18163i −0.803451 0.463873i 0
93.5 2.29475 1.32488i −0.335680 + 1.25278i 2.51060 4.34849i 0 0.889471 + 3.31955i −0.0561740 + 0.0972962i 8.00544i 1.14131 + 0.658935i 0
232.1 −1.80821 + 1.04397i 2.66159 + 0.713171i 1.17974 2.04338i 0 −5.55724 + 1.48906i 1.45563 2.52122i 0.750585i 3.97738 + 2.29634i 0
232.2 −1.37242 + 0.792369i −0.190588 0.0510678i 0.255697 0.442881i 0 0.302032 0.0809291i −0.274164 + 0.474866i 2.35905i −2.56436 1.48053i 0
232.3 0.113711 0.0656513i −0.332179 0.0890070i −0.991380 + 1.71712i 0 −0.0436159 + 0.0116869i −1.39069 + 2.40874i 0.522947i −2.49566 1.44087i 0
232.4 0.885613 0.511309i −2.69193 0.721300i −0.477126 + 0.826407i 0 −2.75281 + 0.737614i 0.481787 0.834479i 3.02107i 4.12812 + 2.38337i 0
232.5 1.94926 1.12540i 1.91913 + 0.514229i 1.53307 2.65535i 0 4.31958 1.15743i −0.638592 + 1.10607i 2.39966i 0.820542 + 0.473740i 0
318.1 −1.80821 1.04397i 2.66159 0.713171i 1.17974 + 2.04338i 0 −5.55724 1.48906i 1.45563 + 2.52122i 0.750585i 3.97738 2.29634i 0
318.2 −1.37242 0.792369i −0.190588 + 0.0510678i 0.255697 + 0.442881i 0 0.302032 + 0.0809291i −0.274164 0.474866i 2.35905i −2.56436 + 1.48053i 0
318.3 0.113711 + 0.0656513i −0.332179 + 0.0890070i −0.991380 1.71712i 0 −0.0436159 0.0116869i −1.39069 2.40874i 0.522947i −2.49566 + 1.44087i 0
318.4 0.885613 + 0.511309i −2.69193 + 0.721300i −0.477126 0.826407i 0 −2.75281 0.737614i 0.481787 + 0.834479i 3.02107i 4.12812 2.38337i 0
318.5 1.94926 + 1.12540i 1.91913 0.514229i 1.53307 + 2.65535i 0 4.31958 + 1.15743i −0.638592 1.10607i 2.39966i 0.820542 0.473740i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.2.x.b 20
5.b even 2 1 65.2.t.a yes 20
5.c odd 4 1 65.2.o.a 20
5.c odd 4 1 325.2.s.b 20
13.f odd 12 1 325.2.s.b 20
15.d odd 2 1 585.2.dp.a 20
15.e even 4 1 585.2.cf.a 20
65.d even 2 1 845.2.t.g 20
65.f even 4 1 845.2.t.e 20
65.g odd 4 1 845.2.o.e 20
65.g odd 4 1 845.2.o.f 20
65.h odd 4 1 845.2.o.g 20
65.k even 4 1 845.2.t.f 20
65.l even 6 1 845.2.f.d 20
65.l even 6 1 845.2.t.e 20
65.n even 6 1 845.2.f.e 20
65.n even 6 1 845.2.t.f 20
65.o even 12 1 65.2.t.a yes 20
65.o even 12 1 845.2.f.e 20
65.q odd 12 1 845.2.k.e 20
65.q odd 12 1 845.2.o.e 20
65.r odd 12 1 845.2.k.d 20
65.r odd 12 1 845.2.o.f 20
65.s odd 12 1 65.2.o.a 20
65.s odd 12 1 845.2.k.d 20
65.s odd 12 1 845.2.k.e 20
65.s odd 12 1 845.2.o.g 20
65.t even 12 1 inner 325.2.x.b 20
65.t even 12 1 845.2.f.d 20
65.t even 12 1 845.2.t.g 20
195.bh even 12 1 585.2.cf.a 20
195.bn odd 12 1 585.2.dp.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.2.o.a 20 5.c odd 4 1
65.2.o.a 20 65.s odd 12 1
65.2.t.a yes 20 5.b even 2 1
65.2.t.a yes 20 65.o even 12 1
325.2.s.b 20 5.c odd 4 1
325.2.s.b 20 13.f odd 12 1
325.2.x.b 20 1.a even 1 1 trivial
325.2.x.b 20 65.t even 12 1 inner
585.2.cf.a 20 15.e even 4 1
585.2.cf.a 20 195.bh even 12 1
585.2.dp.a 20 15.d odd 2 1
585.2.dp.a 20 195.bn odd 12 1
845.2.f.d 20 65.l even 6 1
845.2.f.d 20 65.t even 12 1
845.2.f.e 20 65.n even 6 1
845.2.f.e 20 65.o even 12 1
845.2.k.d 20 65.r odd 12 1
845.2.k.d 20 65.s odd 12 1
845.2.k.e 20 65.q odd 12 1
845.2.k.e 20 65.s odd 12 1
845.2.o.e 20 65.g odd 4 1
845.2.o.e 20 65.q odd 12 1
845.2.o.f 20 65.g odd 4 1
845.2.o.f 20 65.r odd 12 1
845.2.o.g 20 65.h odd 4 1
845.2.o.g 20 65.s odd 12 1
845.2.t.e 20 65.f even 4 1
845.2.t.e 20 65.l even 6 1
845.2.t.f 20 65.k even 4 1
845.2.t.f 20 65.n even 6 1
845.2.t.g 20 65.d even 2 1
845.2.t.g 20 65.t even 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{20} - 6 T_{2}^{19} + 5 T_{2}^{18} + 42 T_{2}^{17} - 66 T_{2}^{16} - 228 T_{2}^{15} + 557 T_{2}^{14} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} - 6 T^{19} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{20} - 2 T^{19} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{20} \) Copy content Toggle raw display
$7$ \( T^{20} - 2 T^{19} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{20} + 16 T^{19} + \cdots + 256 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 137858491849 \) Copy content Toggle raw display
$17$ \( T^{20} + 4 T^{19} + \cdots + 1168561 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 1583721616 \) Copy content Toggle raw display
$23$ \( T^{20} - 10 T^{19} + \cdots + 144 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 206213167449 \) Copy content Toggle raw display
$31$ \( T^{20} - 104 T^{17} + \cdots + 2166784 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 4508182449 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 3748255729 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 1370772640000 \) Copy content Toggle raw display
$47$ \( (T^{10} - 20 T^{9} + \cdots - 28416)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 2978634160384 \) Copy content Toggle raw display
$59$ \( T^{20} + 16 T^{19} + \cdots + 33856 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 826457355409 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 15478905336976 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 64\!\cdots\!24 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 75\!\cdots\!36 \) Copy content Toggle raw display
$83$ \( (T^{10} + 24 T^{9} + \cdots + 3393024)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 329648222500 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 28\!\cdots\!36 \) Copy content Toggle raw display
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